Asymptotic Formula for Biregular Overpartitions

Let ${\bar{p}}_{s,t}(N)$ be the number of overpartitions of $N$ into parts that are not divisible by $s$ or $t$. In this paper, we obtain asymptotic formulas for ${\bar{p}}_{s,t}(N)$ for relatively prime integers $s>t\ge 10$ via the saddle-point method. Our results cover different ranges of the parameters $s$ and $t$ relative to $N$. As an application, we show that ${\bar{p}}_{s,t}(N)$ satisfies higher-order Turán inequalities for all sufficiently large $N$ by using a result of Griffin, Ono, Rolen, and Zagier.